MOTION OF A PARTICLE 137 2. Show that the ratio of the distances described by a falling body 2 ft _ i during the (n — l)th and the nih seconds is - — — - • 2t 71 "T~ 1 3. A juggler keeps three balls going in one hand, so that at any instant two are in the air and one in his hand. Find the tune during which a ball stays in his hand; each ball rises to a height h. 4. Find the shortest time in which a mass m can be raised to a height h-by means of a rope which can bear a tension T. 5. A train passes another on a parallel track. When the two locomotives are abreast one of the trains has a velocity of 20 miles per hour and an acceleration of 3 — '- . while the other has a velocity of 40 miles sec.2 per hour and an acceleration of 1 — '— . How soon will they be abreast sec. again, and how far will they have gone in the meantime? 6. A mass of 1 kg. is hanging from a spring balance in an elevator. After the elevator starts the balance reads 1100 gm. Assuming the acceleration of the elevator to be constant, find the distance moved in 5 seconds. 7. A smooth inclined plane of mass m and inclination a stands with its base on a smooth horizontal plane. What horizontal force must be applied to the plane in order that a particle placed on the plane simultaneously with the beginning of action of the force may be in contact with the plane yet fall vertically down as if the inclined plane were not there?" 8. The pull of a train exceeds the resisting forces by 0.02 of the weight of the train. When the brakes are on full the resisting forces equal 0.1 of the weight. Find the least time in which the train can travel between two stopping stations 5 miles apart, the tracks being level. 9. Give a construction for finding the line of quickest descent from a point to a circle in the same vertical plane. 10. A mass mi falling vertically draws a mass m2 up a smooth inclined plane which makes an angle of 30° with the horizon. The masses are connected with a string which passes over a small smooth pulley at the top of the plane. Find the ratio of the masses which will make the acceleration j • 11. A particle is projected up an inclined plane which makes an angle: a. with the horizon. If Tl is the time of ascent, !T2 the time of descent^ and 4> the angle of friction, show that sin (a - <ft) _ " /!Z\\2 = \TJ